Employment Effects of a Minimum Wage: A Density Discontinuity Design Revisited* Joseph J. Doyle Jr. MIT Sloan School of Management.

Transcription

1 Employment Effects of a Minimum Wage: A Density Discontinuity Design Revisited* Joseph J. Doyle Jr. MIT Sloan School of Management April 005 Abstract: In a classic paper, Meyer and Wise (983) estimated the employment effects of a minimum wage by examining its distortion on the wage distribution, but the approach has been criticized for its reliance on functional form assumptions about the underlying wage distribution. This paper relaxes these assumptions by applying methods commonly used in regression discontinuity design problems to this density discontinuity design: the wage distribution is assumed to be smooth in the absence of a minimum wage and the density is estimated above and below the minimum wage with a local linear kernel density estimator. Meyer and Wise estimate that thirty to fifty percent of young, hourly wage earners who would have earned wages below the minimum can no longer find employment in 978, while the estimates here are closer to ten percent for the same time period, and forty to seventy percent for 985 to 000. The results suggest that large disemployment effects are not driven solely by functional form assumptions, and that the effects appear to grow over time as compliance with the minimum wage increases. *I would like to thank Mark Duggan, Michael Greenstone, Steve Levitt, Roberto Rigobon, and especially Victor Aguirregabiria for helpful comments and ideas. This research was assisted by fellowships from the Social Science Research Council Program on Applied Economics. All remaining errors are my own.

2 . Introduction There is a debate over the effect of a minimum wage on employment (see, for example, Card and Krueger (995), Neumark and Wascher (995), Card and Krueger (998), and Abowd, Kramarz, and Margolis (999)). In a competitive market for lowskilled labor, all workers with productivity levels associated with sub-minimum wages would no longer find employment, and businesses may further substitute away from lowskilled labor. If the labor market is not competitive, employment could rise following the introduction of a minimum wage. Card and Krueger (995) review the empirical literature and argue that increases in the minimum wage tend to have small effects on employment. An earlier study by Meyer and Wise (983a, 983b) found substantial disemployment effects, with an estimated thirty to fifty percent of those who would have earned sub-minimum wages no longer finding employment, and a reduction in employment for all young men of roughly 7%. The idea in their paper was to consider the discontinuity in the wage distribution caused by the minimum wage to identify the employment effects from a single cross section: the more the minimum wage distorted the wage distribution, the greater the estimated effect on employment. Estimates of the size of the distortion rely on some information about the counterfactual wage distribution in the absence of a minimum wage. For example, Meyer and Wise focus on a log normal wage distribution. This strategy has been criticized, as the results are thought to be sensitive to the functional form chosen (Brown, Gilroy and Kohen (98); Card and Krueger (995)). Dickens, Machin, and Manning

3 (994) argue that the main reason the approach is not more widely used is because the estimate depends on the functional form of the wage distribution. This paper relaxes the distributional assumptions by applying the techniques commonly used in regression discontinuity designs to this density discontinuity design problem. In particular, the wage density is estimated above and below the minimum wage using a local linear density estimator. Substituting the functional form assumption with an assumption that the counterfactual wage distribution is continuous, it is then possible to test the effect of the minimum wage on employment and wages. Using a 978 sample of young hourly wage earners similar to Meyer and Wise, only ten percent of sub-minimum wage workers are estimated to be unemployed due to the minimum wage. The disemployment effects are found to be much larger in later years, however, ranging from forty to seventy percent. In general, the disemployment effects tend to be smaller in years when the minimum wage is increased, such as the late 970s. These increases are associated with a rise in the fraction of workers earning wages below the minimum wage. As compliance with the minimum wage appears to increase over time, the estimated disemployment effects also increase. Similarly, some groups that are more likely to receive the minimum wage have smaller disemployment effects. These groups may be more willing to accept wages at or below the minimum. Overall, these results suggest that large disemployment effects are not driven solely by functional form assumptions. The approach also provides a way to estimate the counterfactual wage distribution, and the effect of the reduction in the number of low wage workers on the wage distribution can be examined as well. The median wage is found to increase by 4-

4 % over 985 to 000 when disemployment effects are large, while median wages are naturally unchanged when employment effects are small. The effect of the minimum on wage inequality is particularly pronounced. While the 0 th percentile of young, hourly wage earners is the minimum wage, it is substantially less without the minimum wage. In particular, the median worker earns wages twenty to forty percent higher than the 0 th percentile worker, but without a minimum wage the differences are estimated to increase to a range of forty to seventy percent. The paper is organized into four sections. Section presents the empirical strategy to estimate the disemployment effect. Section 3 describes the Current Population Survey data used in the analysis. Section four discusses the results using all young workers and across subgroups. Robustness checks are provided, including a simulation that suggests the results are not driven by sub-minimum wage reports that stem solely from measurement error. Section five concludes.. Empirical Strategy The empirical strategy relates the observed wage distribution to a counterfactual distribution that would occur in the absence of a minimum wage. The idea is that discontinuity in the wage distribution provides information about the change in employment. Two main assumptions are used to relate the observed density, h ( ), to the counterfactual density, f ( ). Let w represent the wage and M represent the minimum wage: (A-) No spillover effects: For w>m, f ( w) h( w) =, where D (defined below) D rescales the density to take into account the decreased number of workers, if any. 3

5 (A-) For w<m there are 3 cases: () with probability P, the wage continues to be observed (due to non-compliance or non-coverage); () with probability P, a person will be observed with a wage at the minimum, w=m; and 3) with probability -P-P, a person will become unemployed. With these assumptions the observed pdf can be written as: ( ) P f ( w) D P F( M ) h( w) = D f ( w) D if w < M if w = M if w > M where M is a small interval around the minimum wage, and D= F(M)( P P). D is the fraction of workers who continue to be employed after the minimum wage is introduced, and the re-scaling ensures that h ( ) integrates to one. Figure shows the change in the wage distribution: the observed density will be lower than the counterfactual density below M, have a spike at M, and will be higher than the counterfactual density above M (due to the rescaling when D<). The presence of the spike at the minimum wage, and a compression of the wage density, is consistent with the empirical evidence as shown by Dinardo, Fortin, and Lemieux (996). The disemployment effect is given by F(M)( P P): the fraction of workers who lose their jobs. These effects will be small when F(M) is small the minimum is not binding because almost all workers earn more than the minimum or if P or P is close to one. If P= then the law is not binding as workers continue to receive the sub- 4

6 minimum wages. If P= then all workers who previously made less than the minimum are employed with wages equal to the minimum and no one becomes unemployed as a result. To estimate the model, Meyer and Wise imposed an additional assumption: (A-3) f ( ) is lognormal. Box-Cox transformations were tested as well, and the logarithm of wages was found to fit the data best. Nevertheless, the chief criticism of the approach is the reliance on functional form assumptions about the wage distribution. For example, Dickens, Machin, and Manning (994) consider a range of functional forms and find the estimates sensitive to the choice. This paper relaxes the functional form assumption and relies on a continuity assumption that is implicit in the approach: (A-3 ) lime 0 f ( M e) = lime 0 f ( M + e) This assumption makes use of the original idea that the minimum wage creates an artificial distortion in the wage distribution, while the underlying wage distribution is likely to be smooth. It is unlikely that the distribution of worker productivities jumps at a level like $3.35 per hour the minimum wage from Using assumptions (A-), (A-), and (A-3 ), F(M), P, and P can be estimated. First, consider the wage distribution after the minimum wage is in effect. A fraction will continue to receive a wage below the minimum, while a fraction will earn exactly the minimum. These fractions can be expressed in terms of the counterfactual distribution by using (): 5

7 () and P = F( M ) D P = F( M ) D Substituting and for P and P in (), the observed pdf can be rewritten: ( 3) h(w) f(w) F(M) = F(M) f(w) if w < M if w = M if w > M The first and third terms re-weight of the wage density, using the fraction of the observed workers who earn wages above or below the minimum. These weights do not vary with the wage, resulting in a simple likelihood function. Letting N equal the number of individuals with wages below the minimum wage, N equal the number receiving the minimum wage, and N 3 equal the number earning more than the minimum wage. The log-likelihood can then be written as: (4) log L = + + i= N i= N N 3 i= log log F( M ) + log f ( w ) log log( ) log( F( M )) + log f ( w ) i i Letting N equal the total number of individuals in the sample, the maximum likelihood estimates of these proportions are simply the sample proportions: 6

8 (5) MLE = N N (6) MLE = N N The estimation of P is derived from the description of the wage distribution in (): (7) h( M e) = P D f ( M e) ( 8) h ( M + e) = f ( M + e) D Dividing these two expressions and using the continuity assumption (A-3 ), (9) P lim = lim e 0 e 0 h( M h( M e) + e) P can be estimated using a kernel density estimator for h (w) above and below the minimum wage, though estimates close to a boundary require special consideration described below. Recall that P is the probability that a sub-minimum wage worker keeps a job earning below the minimum wage. If the observed wage distribution were continuous implying no distortion from the minimum wage the estimate for P would be one. Also, the estimate of P declines as fewer workers are observed earning wages below the minimum relative to those earning wages above the minimum. In an alternative model, P may be expected to vary with the wage level those earning far less than the minimum wage may have a different probability of remaining at a low wage compared to those earning wages just below the minimum. The estimator here considers the P at the margin of the minimum wage, as made explicit in (7). Meyer 7

9 8 and Wise estimate that the probability of job loss falls as the wage increases toward the minimum wage, suggesting that the disemployment effects estimated here are conservative in terms of the P estimated here. The estimation of P can be derived by dividing the expressions for and in (): = = * or, (0) P P P P Last, the estimation of F(M) can be derived from the description of the observed wage distribution in (3): ) ( ) ( ) ( () ) ( ) ( ) ( () e M f M F e M h e M f M F e M h + = + = Dividing () and () and again using the continuity assumption (A-3 ), F(M) can be solved as: 0 0 ) ( ) ( ) ( lim ) ( lim ) ( (3) + = + + = P e M h e M h M F e e With estimates for F(M), P, and P, the disemployment effects are F(M) ( P P). Card and Krueger (995) argue that another shortcoming of the approach is the restriction of employment effects to be zero or negative. The estimation here does not

10 restrict - P - P to be positive. In particular, if the minimum wage were associated with increased employment at the minimum wage, the estimated would be larger. The disemployment effect is decreasing in, and can be negative as the estimate of P rises. A negative estimate would therefore suggest job gains. Spillover Effects Another criticism of the approach is the reliance on the lack of spillover effects. That is, this method assumes that there is no effect of the minimum wage on wage earners who earn above the minimum. Card and Krueger (995) find some evidence of wage increases for workers just above the minimum wage. Gramlich (976) also finds spillover effects in the US using time series data on average wages, while Benjamin (996) finds little spillover effects for Canada. Green and Paarsch (996) consider variation across time and provinces in Canada and focus on teenage workers. Using a proportional hazard approach (estimating the probability of earning $x given that the person earns at least $x) they find that the minimum wage is associated with a reduction in the frequency of wages just above the minimum for men, and no spillover effects for women. The authors argue that this may be due to substitution from low wage workers to higher paid workers. In addition, they argue that young workers have such high turnover that when they are hired the minimum wage acts as a focal wage. If these estimates apply to young workers in the US, then these reverse spillover effects would result in estimates closer to zero, or job gains. Local Linear Density Estimation The identification hinges on estimating the density about the boundary M. In 990, the estimated disemployment effect is 5% and is 3.8%. To achieve negative disemployment effects, would have to increase to over %. 9

11 Kernel estimation relies on smoothing density estimates, but within one bandwidth, h, of the boundary, there is a smaller interval to carry out the estimation. This results in a bias of order h near the boundary which is worse than the usual bias of order h in the interior. The problem is addressed in a kernel regression setting by using a local linear regression estimator (Fan, 99; Hahn, Todd, and Van der Klaauw, 999). In an analogous way, Jones (993) shows that local linear density estimation can be used to address the boundary problem. In particular, a and b are chosen to minimize: ( w u) ( 3) h K n h { f ( u) a b( w u) } du where f n is the empirical density function. Jones shows that away from the boundary this is the usual kernel density estimator. Within h of the boundary, an explicit solution to (3) can be derived where the kernel is automatically adjusted as it reaches the boundary. Consider a boundary located at zero and positive data. Letting p=x/h, the boundary kernel K L (x) that results is: a ( p) a( p) x ( 4) K L ( x) = K( x) a ( p) a ( p) a ( p) where 0 ( 5) a ( p) = l p l u K( u) du For example, the expression using the normal kernel is: Φ( p) + ( x p) φ( p) ( 6) K L ( x) = φ( x) Φ( p) φ ( p) ( Φ( p) pφ( p) ) When the boundary is from above, the terms in the numerator are subtracted instead of added. In the present study the main estimates of interest are at the boundary where p=0, 0

12 though estimates of the effect of the minimum on the entire distribution will be discussed as well. At the boundary, the expression simplifies to: Φ(0) + xφ(0) ( 7) K L ( x) = φ( x) = x φ( x) (0) φ (0) + Φ In the Current Population Survey data used in the analysis, the sampling weights can be incorporated. The appendix shows the derivation of the resulting kernel density estimator with weight θ i : n ( 8) h( w) = h θ i i= K L w wi h Figure shows the kernel density estimates with and without the boundary correction for non-student wage earners ages 6-4 in 000. The correction results in higher estimates both below and above the minimum wage. This is found in earlier years as well, with the estimated maximum often at the minimum wage itself. Given the no spillover assumption, this implies that the estimated counterfactual distribution will also have a maximum at the minimum wage for those years. While it is possible that this young sample does have a maximum near the minimum wage, it suggests the possibility of an influence of the minimum wage on the counterfactual. Nevertheless, the estimates with the boundary correction tend to be more conservative as discussed below. A second issue is the choice of a bandwidth. There are many methods of choosing a local bandwidth near the boundary (Muller, 99). Zhang and Karunamuni (998) found that when estimating a function at an endpoint, the optimal bandwidth choice is twice the bandwidth that would be optimal for interior points. Intuitively, increasing the size of the The bandwidth is $.03 above the minimum wage and $0.67 below the minimum wage. As described below, this is a rule of thumb bandwidth for estimation at the boundary the primary interest in this paper.

13 bandwidth at the boundary allows more observations to be used to estimate the density. The main results will use two times Silverman s rule of thumb and the estimates are tested for sensitivity to the choice of bandwidth. 3. Data Description Meyer and Wise used the May 978 supplement to the Current Population Survey, which recorded hourly wages. This paper also uses the May 978 data for comparability, and also estimates the employment effects for using the outgoing rotation groups for the CPS. These later datasets allow the use of all twelve months of the data. To concentrate on the groups most likely to be affected by the minimum wage and to compare the results to earlier estimates, the sample is defined in the same way as Meyer and Wise (983b). 3 First, the focus is on young workers, those aged 6-4. Results will be presented separately for those aged Second, given the young age, students are excluded. 4 Third, the individual had to report an hourly wage. Last, in later years, up to twelve states had minimum wages greater than the federal minimum wage. These states are dropped from the later analyses to estimate the single discontinuity created at the federal minimum wage. These restrictions result in 600 observations in 978, and roughly ten to twenty thousand observations for The empirical strategy relies on information about wages in the absence of the minimum wage being provided by actual observations below the minimum. For some groups, this is due to non-coverage although an expansion in January 978 and a smaller 3 Meyer and Wise focus on men, and estimates will be provided for men and women separately below. 4 From , non-students are defined as those whose major activity is either working or looking for work, as opposed to a student. From 994 onward, the variable definitions change and the analysis here considers those who reported not being a full or part-time student.

14 expansion in 985 extended the federal minimum to most workers. For other workers, this could be due to non-compliance. Table shows that a substantial fraction of young workers report wages below the minimum wage, ranging from 3% to 9% from This tends to increase in years when the minimum wage is increased, such as the late 970s, the early 990s and the late 990s. Meanwhile, the fraction of young, hourly wage workers receiving the minimum wage is highest in the late 970s at 4%, falling to 6% by 996 and only.4% in 000. To explore the types of observations below, at, and above the minimum wage, Table provides summary statistics for 990. Those at or below the minimum wage tend to be female, work part-time, and hold less education compared to hourly wage earners above the minimum, suggesting that the sub-minimum wage reports are not simply measurement error. Meanwhile, workers at the minimum wage are more likely to be black and from the South. Estimates will be presented for these groups separately. Those below the minimum wage are most likely to work in the retail sector, especially in restaurants (53%). This is partly due to the allowance of wages below the minimum for tipped workers. These workers provide some information about low-wage work, though in these cases the employers are able to attract workers with competitive compensation due to the tips. In addition, a substantial fraction of restaurant workers earn exactly the minimum wage, and the effect of the minimum wage change on this group may be large. 5 Results will also be shown separately by industry, and for a model that excludes restaurant workers. 5 To the extent that restaurant workers have a different minimum wage, the estimates here have the advantage of focusing on the wage distribution just above and below the federal minimum. Currently, the federal minimum is $5.5 and the minimum wage for tipped workers is $.3 per hour. 3

15 4. Results The results suggest that the minimum wage is associated with large disemployment effects from the mid-980s through 000, though effects tend to be smaller in years when the minimum wage is increased, especially the late 970s and early 980s. Table 3 includes the estimates for 978 and five year intervals from 980 to 000 (see the appendix for all years). In 978, the same year as Meyer and Wise (983b), the probability that a sub-minimum wage earner continues to work at wages below the minimum is 0.3, compared to the Meyer and Wise estimate of 0.3. P, the probability that a wage earner receives a raise up to the minimum is 0.59, compared to 0.34 found previously. The resulting estimate of the probability of job loss is therefore smaller using the kernel density approach: 0%, with a disemployment effect for all young, hourly wage earners F(M)*(- P-P) of.% (s.e.=.6%). 6 The estimate for 980 suggests job gains when the minimum wage was increased to $3.0 with a.6% increase in employment (s.e.=0.9%). Later years show larger disemployment effects. Table 3 shows that the probability of job loss is generally close to 50-60%, with overall disemployment effects between 8 and 9%. Standard errors tend to be small, ranging from -%. These results suggest that large disemployment effects can be found when the functional form assumption is relaxed. From , the nominal minimum wage is constant at $3.35 following a series of increases in the late 970s and early 980s. Over time, fewer workers are observed in non-compliance and at the minimum wage, and more sub-minimum wage 6 Three hundred samples were re-drawn to calculate the bootstrapped standard errors. 4

16 workers are estimated to lose their jobs. The result suggests that the disemployment effects may grow as non-compliance becomes more difficult over time. This effect tends to be found in the 990s as well. For example, the minimum wage increased in April 990 and April 99 to $3.80 and $4.5. When the estimation uses the ten months from May to March that include the new, higher minimum wage, the fraction with wages below the minimum increases from three percent to six percent, and the disemployment effects decline to 3-5%. The minimum wage was increased again in October 996 and September 997. In the months after these changes, the fraction that reports sub-minimum wages also increases, resulting in a larger estimate of P. The associated disemployment effect in 997 continues to be large (8%), but the effect in 998 is estimated to be only 3.%. These results are consistent with higher non-compliance at the start of a new minimum wage regime, leading to smaller disemployment effects in the short-run. To consider the effect of the minimum wage on the wage distribution, the counterfactual can be estimated using (3), with results similar to Figure. Figure 3 replicates the kernel density estimation shown above, and includes the estimated counterfactual density as well. The counterfactual is above the observed density in the range of wages less than the minimum wage, and below the observed density at wages higher than the minimum, by construction when the estimated disemployment effect is positive. Estimated densities for are considered below. Effects across Sub-groups The relationship between the likelihood of receiving a low wage and the disemployment effects may be positive or negative. The minimum wage is generally 5

17 thought to disadvantage those at the lowest wages, though previous estimates do not find this to be the case in general. One explanation is that the lowest wage workers may also be the most likely to accept sub-minimum wages, thereby avoiding the regulation though non-compliance. To examine this relationship, the effects of the minimum wage are compared across demographic groups. The Meyer and Wise estimates focused on men, and Table 4 shows that the estimated disemployment effects are lower for that group. Job gains are found in 978 & 980, while the probability-of-job-loss estimates range from 40%-60% for , with overall disemployment of 7-0%. Women are roughly twice as likely to report wages below the minimum wage and also more likely to receive the minimum wage. The disemployment effects are larger as well: a range of 6% to 37%. Meanwhile, hourly wage earners aged 5-34 are less likely to receive wages at or below the minimum wage, reflecting their greater in experience. Nevertheless, estimates show larger disemployment effects in 978 and 980, and continue to find effects in , albeit smaller than those aged 6-4. For wage earners below the minimum wage, the fraction not finding a job is actually higher in each time period compared to younger workers, consistent with earlier estimates by Meyer and Wise. Older workers making sub-minimum wages may have worse attributes compared to the teenage group, which may result in worse employment prospects after the minimum wage. Across industries the positive relationship between minimum wage exposure and disemployment effects are particularly strong. Table 5 shows that manufacturing industries tend to have the fewest hourly workers at or below the minimum wage, followed by service industries, while retail industries have the greatest exposure. The 6

18 disemployment effects are found to be small for the manufacturing sector and the largest for the retail sector (-39%). To further explore this relationship, Table 6 reports the results for subgroups that are more likely to receive wages at or below the minimum wage. Panel A shows that part-time workers are particularly more likely to receive wages at and below the minimum, and large job gains are estimated in 978 and 980, possibly due to employers substituting full-time workers with part-time workers. This effect is not found in later years, however, with relatively larger disemployment effects found for part-time workers. A similar pattern is found for those who live in the South. These hourly wage earners are more likely to receive the minimum wage, and small job gains are estimated for 978, no effect is found in 980, and disemployment effects are slightly larger in the later years compared to those who live in other regions of the country. Apart from women, a negative relationship between personal characteristics that are associated with greater minimum wage exposure and disemployment effects is found. Table 6 shows that high school dropouts and blacks are more likely to receive wages at the minimum wage, yet the disemployment effects for these groups are smaller. In particular, young black wage earners have a larger fraction at the minimum wage, leading to larger estimates of P. High school dropouts are more likely to have wages at or below the minimum wage leading to higher estimates of both P and P. Groups that are more likely to accept sub-minimum wages or the minimum wage may have smaller disemployment effects of the minimum wage as they successfully compete with other groups for the minimum wage positions or avoid the regulation altogether. Robustness 7

19 The minimum wage law attempts to prohibit the very wages used to estimate the model. This suggests that the approach may rely on measurement error to observe these sub-minimum wages. One test reported in Table 7 considers estimates from a model where the sub-minimum wage observations were constructed to be the result of measurement error. All observations below the minimum wage were excluded, and a noise term was added to each of the wage observations. This term is normally distributed with mean zero and a standard deviation chosen so that the fraction of observations below the minimum wage matches the original dataset. For example, in 980 the standard deviation is forty-five cents. 7 In addition, a fraction of observations at the minimum wage, equal to the fraction of observations below the minimum wage in the original sample, was excluded so that the fraction observed at the minimum wage was again similar to the original sample. Table 7 shows that the estimated P is much higher with a range of 0.46 to 0.8. P is derived from this P estimate and is larger as well. The resulting disemployment effects are all negative, with larger magnitudes for 978 and 980. These estimates essentially consider the shape of the wage distribution above the minimum wage, with the measurement error reflecting these observations across the boundary at the minimum wage. The results suggest that sub-minimum wage reports that are due solely to measurement error are unlikely to result in large disemployment effects. Table 8 shows that the results are similar across a number of robustness checks. First, Figure showed that the local linear kernel density estimation tended to increase the estimates of the density both above and below the minimum wage. The estimated maximum of the wage density is often at the federal minimum wage, resulting in a maximum of the counterfactual wage at the federal minimum as well. To the extent that 7 The standard deviations chosen for 978 through 000 were: 0.47, 0.45, 0.50, and

20 the maximum would be somewhat larger than the minimum wage in its absence, the estimated P here may be an understatement of the true P. Alternatively, the wages just below the minimum wage generally show a slight decline just below the minimum, and the local linear correction increases the estimates there. If this effect dominates, then the estimated P will be larger with the boundary correction. Kernel density estimates using the same bandwidths and a normal kernel, but no local linear correction term, result in smaller estimates of P and larger estimated disemployment effects as shown in Table 8. The local linear correction appears to result in more conservative estimates of the disemployment effects. A second issue suggested by Table is that restaurant workers likely play an important role. Those who receive tips generally are allowed to receive wages below the minimum wage. That said, the effect of the minimum wage on employment for this group may be substantial, as many of these workers earn exactly the minimum wage as well. Table 8 shows that the disemployment effects are smaller when this group is excluded from the analysis, though the pattern of results is similar. For 978 and 980, the effects are small or imply small job gains. For , the estimated probability of job loss for those who would have earned wages below the minimum range from 40% to 6%, and overall disemployment effects are between 4 and percent. Third, the main results estimate the density one cent above and below the minimum wage. Wages tend not vary at the one-cent level, however, and this can lead to smaller estimates of the density just below and above the minimum wage. When the density is estimated five cents above and below the minimum wage instead, similar results are found. Two years that are sensitive to the interval choice are 979 and 980, 9

21 when the nominal minimum wages were $.90 and $3.0, respectively. With a large number of observations at $3.00, this tends to decrease the estimated P for 979, and increase the estimate for 980. When an eleven cent interval is used to remove the observations at $3.00 from entering into the estimation, the estimated disemployment effect is smaller in 979 (6%), and becomes large and positive for 980 (7%). For this reason, the 980 job gain estimates should be interpreted with some caution. It does not appear to be entirely due to the proximity of the $3 wage, however, as job gains are also estimated in 98 the first year the minimum wage increased to $3.35. When the results are compared across years, one issue that arises is that the mix of states is different those that have the federal minimum at the time. Panel D shows that the results are slightly higher when the set of states is the same throughout. A final robustness check tests the effect of the bandwidth choice. Recall that a larger bandwidth is thought to be appropriate at a boundary where the number of observations in the smoothing window is allowed to increase. In 980, for example, the bandwidth calculated with Silverman s rule of thumb is thirty cents above the minimum wage and fifteen cents below the minimum wage, and the main results of density estimates described above used bandwidths twice as large. 8 Table 9 shows that when the bandwidth is set equal to Silverman s rule of thumb, the estimates are similar, although no disemployment effect is found for 000. When the bandwidth is increased to twice that used in the main results, the disemployment effects are found to be larger, especially in 000. Again, the bandwidth chosen in the main results is generally a conservative choice for the disemployment effect estimates. 8 Bandwidths used in 978, 980, 985, 990, 995, and 000 above the minimum wage were 70, 60, 80, 88, 96, 03, respectively, while below the minimum wage the bandwidths were 30, 30, 39, 4, 46, and 67, all in nominal dollars. 0

22 Effects on Wages While the minimum wage may lower employment, it is expected to raise the wage of those who are employed at low wages. In a model with no spillover effects, the median wage will also increase if sub-minimum wage workers are less likely to be employed. Figure 3 showed that the minimum wage does compress the wage distribution at the lower end for young, hourly wage earners in 000. Figure 4 shows estimates of the observed and counterfactual wage distributions above and below the minimum wage for , ignoring the mass at the minimum wage. In 978, the disemployment effect is small, so the observed and counterfactual distributions above the minimum wage are nearly identical. In 980, with job gains predicted, the counterfactual distribution is actually slightly larger than the actual distribution above the minimum wage. The estimated maximum tends to be at the minimum wage, though this is partly due to the boundary correction. While this correction results in smaller disemployment effects, it may increase measures of wage inequality as more mass is estimated just above and below the minimum wage. Table 0 reports percentile estimates of the actual and counterfactual wage distributions, including the mass of observations at the minimum wage in the actual distribution. Consistent with estimates by Meyer and Wise, the effect of the minimum wage on median earnings is small in 978: an increase in the median wage from $9.03 to $9.3 (in 000 dollars). In 980, the median in the counterfactual wage distribution is actually higher than the actual distribution, reflecting the small job gains found for low wage workers. When disemployment effects are larger in 985 to 000, the median wage increases more substantially. A % increase is estimated in 985, and the estimates

23 decrease to 4% by 000. In terms of inequality, the 0 th percentile is at the minimum wage in the observed distribution, and substantially below the minimum wage in the counterfactual distribution. The last three columns show that the difference between the 90 th and the 50 th percentiles are similar in the actual and counterfactual distributions. The difference between the 50 th and the 0 th percentiles is much higher. In 985, where the largest disemployment effects were found, the median wage earner makes 33% more than the 0 th percentile worker with the minimum wage, and 7% more than the 0 th percentile worker in the counterfactual distribution without the minimum wage. This difference declines in subsequent years, to a comparison of 43% versus 55% by 000. These results show that the wage increases, and the reduction in inequality, are greatest when the disemployment effects are also large as some low-wage workers are no longer in the work force. 5. Conclusion A chief criticism of using the distortion of the wage distribution to estimate employment effects of a minimum wage has been its reliance on functional form assumptions. This paper substitutes these assumptions with an assumption that the counterfactual wage distribution in the absence of the minimum wage is continuous. When local linear estimation common in the regression discontinuity design context is applied to this density discontinuity design, large disemployment effects are found for In these years, the probability of job loss for a worker who would have earned a wage below the minimum is between fifty and seventy percent. For young wage earners as a group, the estimated disemployment effect is roughly ten to twenty percent.

24 Following minimum wage increases in the late 970s, and the early and late 990s, the fraction of workers observed below or at the minimum wage tends to fall over time, and the estimated disemployment effects tend to increase. The estimates suggest that as compliance with the federal minimum increases over time, the disemployment effects increase as well. With the exception of women, workers with personal characteristics associated with lower wages tend to have smaller disemployment effects. These workers are more likely to be found earning wages at or below the minimum wage. To the extent that employers substitute toward these workers, these groups may avoid some of the disemployment effects associated with a minimum wage. In fact, job gains for part-time workers are found in some years, suggesting that employers may substitute away from full-time workers in response to the minimum wage. Using the simple re-scaling model, it is also possible to construct the counterfactual wage distribution in the absence of the minimum wage. The disemployment of low-wage workers in 985 to 000 is found to increase median wages by 4-%. Further, the minimum wage is found to compress the wage resulting in significantly smaller measures of inequality. The approach has a number of limitations. First, it relies on sub-minimum wage reports that place a greater emphasis on the potential for measurement error, though no disemployment effect is found in a simulation where all sub-minimum wage observations are due to measurement error by construction. Second, the estimation procedure continues to rely on an assumption of no spillover effects workers earning wages above the minimum wage are assumed not to be affected. To the extent that the minimum wage 3

25 depresses the frequency of wages just above the minimum, the results here may understate the true effect, while the effect may be overstated if the minimum wage is associated with an increase in the frequency of wages just above the minimum. Despite these limitations, the results suggest that large disemployment effects are not solely the result of functional form assumptions. 4

26 References Abowd, J., Kramarz, F., and D. Margolis. (999) Minimum Wages and Employment in France and the United States. NBER Working Paper. No March. Benjamin, D. (996) Minimum Wages in Canada. Manuscript. Department of Economics, University of Toronto. November. Brown, C., Gilroy, C. and A. Kohen. (98) The Effect of the Minimum Wage on Employment and Unemployment. Journal of Economic Literature. Vol Card, D. and A. Kreuger. (995) Myth and Measurement: The New Economics of the Minimum Wage. Princeton, N.J. : Princeton University Press. Card, D. and A. Krueger. (998) A Reanalysis of the Effect of the New Jersey Minimum Wage Increase on the Fast-Food Industry with Representative Payroll Data. NBER Working Paper No Dickens, R., Machin, S. and A. Manning. (994) Estimating the Effect of Minimum Wages on Employment from the Distribution of Wages: A Critical Review. Centre for Economic Performance Discussion Paper. No. 03. August. Dinardo, J., Fortin, N., and T. Lemieux. (996) Labor Market Institutions and the Distribution of Wages, : A Semiparametric Approach. Econometrica. Vol. 64, No. 5. September Fan, J. (99) Design Adaptive Nonparametric Regression. Journal of the American Statistical Association. Vol Gramlich, E.M. (976) Impact of Minimum Wages on Other Wages, Employment and Family Incomes. Brookings Papers on Economic Activity Hahn, J., Todd, P., and W. Van der Klaauw. (993) Evaluating the Effect of an Antidiscrimination Law Using a Regression-Discontinuity Design. NBER Working Paper. No. 73. Jones, M.C. (993) Simple Boundary Correction for Kernel Density Estimation. Statistics and Computing Meyer, R.H. and D.A. Wise (983a). The Effects of the Minimum Wage on Employment and Earnings of Youth. Journal of Labor Economics. Volume, Number Meyer, R.H. and D.A. Wise (983b). Discontinuous Distributions and Missing Persons: The Minimum Wage and Unemployed Youth. Econometrica. Volume 5, Issue 6. November Muller, H. G. (99) Smooth Optimum Kernel Estimators Near Endpoints. Biometrika. Vol Neumark, D. and W. Wascher. (995) The Effect of New Jersey's Minimum Wage Increase on Fast-Food Employment: A Re-Evaluation Using Payroll Records. NBER Working Paper No. 54 Neumark, D., Schweitzer, M. and W. Wascher (998) The Effects of Minimum Wages on the Distribution of Family Incomes: A Non-Parametric Analysis. NBER Working Paper. No April. Zhang, S. and R. J. Karunamuni (998). On Kernel Density Estimation Near Endpoints. Journal of Statistical Planning and Inference. Volume

27 6 Appendix: Incorporating CPS Sampling Weights The CPS sampling weights represent how many people the individual in the survey represents. Let N equal the population of young hourly wage earners in the United States. If we had data on the entire population the kernel estimate would be: ( ) = = N i i L h w w K Nh w h A ) ( ) ( But, instead we have weights. Letting N i equal the weight and n equal the sample size: ( ) = = n i i L i h w w K N Nh w h A ) ( ) ( or = = n i i L i h w w K h w h A ) ( 3) ( θ For subgroups such as individuals who earn more than the minimum wage, N equals the number of such people in the population.

28 Figure : Wage Distribution With & Without Minimum h(w) f(w) M w

29 Figure : 000 Wage Distribution Boundary Corrected No Correction Year-Old Wage Earners, Non Students Wage

30 Figure 3: 000 Observed & Counterfactual Wage Distributions h(w) f(w) Wage 6-4 Year-Old Wage Earners, Non Students

31 Figure 4: Observed and Counterfactual Wage Distributions: 6-4 Year-Old Wage Earners, Figure 4A: 978 Figure 4B: h(w) f(w) h(w) f(w) Real Wage Real Wage Figure 4C: 985 Figure 4D: h(w) f(w) h(w) f(w) Real Wage Real Wage Figure 4E: 995 Figure 4F: h(w) f(w) h(w) f(w) Real Wage Real Wage

32 Table : Observations At or Below the Minimum Wage Non-student, Hourly Wage Earners, Ages 6-4 Federal Minimum Year Nominal Real % Below % At (000$) Data: 978 from May CPS; : Merged Outgoing Monthly Samples. Minimum wage as of May in each year. Estimates weighted by CPS weights. 990 includes May March 99; 99 includes May 99 - March 99; 997 includes Nov. 996 through Aug. 997; 998 includes Oct. 997 through Sept. 998.

33 Table : Descriptive Statisitics Non-student, Hourly Wage Earners, Ages 6-4: 990 Wage < Minimum Wage = Minimum Wage > Minimum Mean Std. Dev Mean Std. Dev Mean Std. Dev Wage (cents) Personal Age Characteristics Male Part-time Race White Black Educational Less than High School Attainment High School Some College College Industry Manufacturing Service Retail Restaurants Other Region Northeast South Midwest West Observations Data: CPS Merged Outgoing Monthly Samples. Estimates weighted by CPS weights.

34 Table 3: Disemployment Estimates Minimum Wage Model Estimates Year Nominal Real Fraction Below Fraction At P(Keep W<Min) P(Raise to Min) P(Job Lost) Disemployment Obs. ($) (000$) P P - P - P F(M) (-P-P) (0.004) (0.005) (0.07) (0.045) (0.066) (0.06) (0.00) (0.003) (0.07) (0.03) (0.044) (0.009) (0.00) (0.003) (0.009) (0.07) (0.035) (0.03) (0.003) (0.00) (0.08) (0.0) (0.07) (0.03) (0.00) (0.00) (0.08) (0.05) (0.04) (0.04) (0.00) (0.00) (0.030) (0.07) (0.044) (0.0) All non-student, hourly wage earners, aged 6-4. Data: 978 from May CPS; : Merged Outgoing Monthly Samples. Minimum wage as of May in each year; 990 includes May March 99. Estimates weighted by CPS weights. Bootstrapped standard errors in parentheses.

35 Minimum Wage Model Estimates Year % Below & At P(Keep W<Min) P(Raise to Min) Disemployment Observations P P F(M) (-P-P) Table 4: Estimates by Sex & Age Group A. Men B. Women C. Ages All non-student, hourly wage earners; The sex comparison is for workers aged 6-4. Data: 978 from May CPS; : Merged Outgoing Monthly Samples. 990 includes May March 99 Estimates weighted by CPS weights.

36 Minimum Wage Model Estimates Year % Below & At P(Keep W<Min) P(Raise to Min) Disemployment Observations P P F(M) (-P-P) Table 5: Estimates by Industry A. Manufacturing B. Service C. Retail D. Other Industries All non-student, hourly wage earners, aged 6-4. Data: 978 from May CPS; : Merged Outgoing Monthly Samples. 990 includes May March 99 Estimates weighted by CPS weights.

37 Minimum Wage Model Estimates Year % Below & At P(Keep W<Min) P(Raise to Min) Disemployment Observations P P F(M) (-P-P) Table 6: Low-Wage Subgroups A. Part-time Workers B. South C. Less than High School D. Black All non-student, hourly wage earners, aged 6-4. Data: 978 from May CPS; : Merged Outgoing Monthly Samples. 990 includes May March 99 Estimates weighted by CPS weights.